Take a white cotton kitchen towel. Dampen a corner of it and rub it good on your red fabric. If the red shows up on your white towel, you know you have a bleeder. If so, I agree with Bear about the
Vicki Welsh method, although I don't think it takes any extra time and effort - in fact, I think it's LESS time and effort than multiple washings with color catchers.
FYI in case you're not familiar with Vicki Welsh, she's a hand-dyer and quilter, and took it upon herself to research and experiment with all the various methods of setting dyes and fixing bleeders, including Retayne, Synthrapol, Dawn, and regular detergent. Her research, data, and conclusions are all in that link above. If you want her instructions on how to do it with your fabrics, click the link at the top of her page where it says "Download the full instructions here".